# The wave of a photon/The double slit experiment

The double slit experiment is especially suitable for showing the wave properties of photons because the wave is split in two slits in two waves, introduce a time (so phase) difference while compared both by interference.

If light passes a single slit (1) it will diffract, resulting in a wide "wave" on the detector. With a second slit (2) there will be two separate waves, which add up at the detector. At places outside the centre of the detector there will different path lengths of both waves causing a phase difference. The sum of both waves will show an interference pattern. With a single photon the effect will be the same: in time the same wave will be build up with single photons. So what the wave shows, is the probability of absorption of a photon at the that position of the detector, according the quantum mechanical formula above.

Formulation: at the detector the two waves add up as: I = sin(ωt-φ/2) + sin(ωt+φ/2), if φ is the phase shift between both waves. Then the probability P = 0.5(1+cosφ)

Another classical formulation in wikipedia.

Energy

If with a single photon is measured, the detector it will show a certain energy. If one slits are narrowed, the interference pattern will change, but not the measured energy. Conclusion:

1. The amount of the wave does influence the absorption probability pattern
2. The amount of the wave does not influence the energy. So the wave does not contain energy in its volume (although its frequency is a measure of the energy)

Spatial coherence

In (2) the waves through both slits cancel each other completely in B, so are equal in size in both slits. Still the particle must have gone though one of the slit. That is by quantum definition (indivisible particle) and it has never been measured to split up in two particles. If detectors are placed just after the slits, they always measure one particle in on or the other slit, while the waves are the same at avery slity. Conclusion:

1. the wave has the same amplitude at a spatial distance of the photon.

Coherence length

In (2) the interference can extend to the left and right over many periods. Suppose in C the phase difference is 10 periods. If the particle went through slit 1, its wave will interfere with the wave which comes 10 periods later out of slit 1. But if the particle went through slit 2, its wave will interfere with the wave which came 10 periods earlier out of slit 1.

1. In direction of propagation (and time) the amplitude is about the same, as much wavelengths before and after the particle as the interference pattern shows. This is the coherence length of the wave.
1. According Feynman with a point source the wave amplitude degrades with 1/r2, with r is the distance to the source.

Wave carrier

With a single slit (1) the wave is differacted, according the Huygens-Fresnel principle. However the principle is calculated for a wave with carrier. It is the interference of every point of the carrier radiating spherical which result in the diffraction pattern. This not caused by the particle because the diffraction behaviour depends on the wavelength and in the double slit the slit which passes only the wave gives the same diffraction as the slit which passes particle + wave. Conclusion:

1. The wave behaves like a wave with carrier (this certainly not a classical carrier, like the ether in the past)

Point source